When people present arguments in writing or speech, they sometimes acknowledge assumptions: "While several of my colleagues in Psychology have argued persuasively against Theory X, in this talk, I assume X is true." It's nice when people are clear about their assumptions, both to themselves and to their audiences. It is also quite rare. In popular media and in articles written for scientists and academics, contestable assumptions are routinely passed over in silence. An important part of improving your ability to read and evaluate argumentative texts is training yourself to detect problematic assumptions in your own and in other people's reasoning.
For example, consider again the first argument presented above: "You shouldn't feel bad that you're going to die because death is inevitable.” What does this argument assume? Plausibly, that you should never feel bad about inevitable things. Perhaps something about this assumption smells fishy to you... It's inevitable (near enough, given the laws of nature) that all sapiens will one day be extinct. Should this recognition make you feel less bad about the possibility of a meteor wiping out humankind forever? I'm inclined to say "no". Having identified an unstated assumption, we are now in a better position to make progress on evaluating the argument.
Unstated assumptions are sometimes also called "implicit claims" to contrast them with claims that an author explicitly makes by writing them down or saying them out loud.